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A method for determining the magnitude of the Raman scattering matrix element for diamond-type crystals
E. Burstein, S. Ganesan
To cite this version:
E. Burstein, S. Ganesan. A method for determining the magnitude of the Raman scatter- ing matrix element for diamond-type crystals. Journal de Physique, 1965, 26 (11), pp.637-638.
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637.
A METHOD FOR DETERMINING THE MAGNITUDE
OF THE RAMAN SCATTERING MATRIX ELEMENT FOR DIAMOND-TYPE CRYSTALS
By E. BURSTEIN (1) and S. GANESAN (2),
Laboratory for Research on the Structure of Matter and Physics Department, University of Pennsylvania, Philadelphia, Pa., U. S. A.
Résumé. 2014 On discute le mécanisme qui produit la bande d’absorption infrarouge du premier
ordre induite par un champ électrique dans les cristaux du type diamant.
L’intensité de cette bande est déterminée par la relation entre la polarisation électrique et le déplacement atomique dans la maille. La mesure de la constante d’absorption de la bande induite devrait donc fournir des renseignements quantitatifs sur les éléments de la matrice de diffusion Raman du premier ordre. Ce phénomène d’induction de bandes infrarouges par le champ élec- trique doit se produire aussi pour des modes actifs en Raman, dans d’autres structures cristallines à centre de symétrie, et pour des modes d’impuretés actifs en Raman.
Abstract. 2014 A discussion is presented of the mechanism for the electric field induced first order infrared absorption band in diamond-type crystals. The strength of the induced band is deter- mined by the dependence of the electronic polarization on the relative atomic displacements in
the unit cell. A measurement of the absorption constant of the induced band should therefore
provide quantitative information about the first order Raman scattering matrix elements. The
phenomenon of field induced infrared absorption bands should also exist for Raman active modes in other centrosymmetric crystal structures, and for Raman active impurity modes.
LE JOURNAL DE PHYSIQUE TOME 26, NOVEMBRE 1965,
1. Introduction. - As part of a general theore-
tical and experimental investigation of "morphic"
effects in crystals induced by electric fields, it was
of interest to us to study the effect of an applied
electric field on the infrared absorption spectrum
of diamond-type crystals. In this type of crystal,
the effective charge of the atoms is zero and there
is no first order (one phonon) resonance absorption
of infrared radiation by the fundamental (q = 0) optical modes. Such crystals do exhibit well defi-
ned, although weak, higher order absorption bands arising from higher order terms in the electric moment [1]. From a phenomenological point of view, the application of an electric field removes
the center of symmetry of the diamond structure,
so that the q = 0 optical vibration modes which
are active in first order scattering also become
active in first order infrared absorption (3). As
shown in Section 2, the strength of the field induced
absorption band is determined by the linear depen-
dence of the static electronic polarizability on the
relative displacements of the two atoms in the
(primitive) unit cell, aoco p X. Since the static and optical frequency electronic polarizabilities are essentially equal in diamond-type crystals (fx(co) = x(0) = mo) a measurement of the strength
(1) Research supported in part by the U. S. Office of Naval Research.
(2) Research supported by the Advanced Research Pro-
jects Agency.
(3) We are also investigating the corresponding morphic
effect in NaCI and CsCI-type crystals in which a first order
Raman spectrum is induced by an applied electric field,
of the induced absorption band should provide quantitative information about the magnitude of
the first order Raman matrix elements which are
determined by è)oc{ (0) IbX.
2. Theoretical. - The mechanism, for the field
induced infrared absorption band may be visua- lized in terms of a simple " atomic " model as
follows : The applied electric field induces a
dipole moment at each atom. The fundamental
optical vibrations of the lattice cause a change in
the electronic polarizability of the atoms so that
the induced dipole moment varies in magnitude
and orientation at the frequency of the lattice
vibrations.
The expressions for the electronic polarizability,
oc, and the induced dipole moment, ti, of the atoms,
take the form :
where a0 is the static (m == 0) electronic polari- zability of the atoms ; and X is relative displa-
cement of the two atoms in each (primitive) unit cell, which varies at the frequency of the q sw 0 optical vibration modes (4).
The first term in the expression for the induced
(4 ) For the present purposes, we neglect the effect of the applied field on the amplitude and frequency of vibration modes.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011063700
638
dipole moment, (1.0 = ao Eo represents the static electric moment. The second term
I ---
represents the time varying electric moment which provides the coupling to the infrared radiation.
The second term can also be expressed as
where
represents the field induced effective charge of the
atoms. We see that ei is proportional to boco/ZX,
the first order dependence of the static electronic polarizability on relative displacement ampli-
tude. Since the magnitude of the infrared absorption constant depends on the square of the electric moment, the strength of the induced absorption band will be proportional to 2 - ? Eo i.e., it will vary as E2
el =
zx 0 I.e., It WI vary as o.
The matrix element for first order Raman scat-
tering depends on the magnitude of boc(co) IbX
where cx(co) is the electronic polarizability of the
atom at the frequency of the exciting radiation.
However, in the case of diamond-type crystals, the
static and optical frequency values of the electronic
polarizability are essentially identical, except when
the excitation frequency is close to a resonance frequency. Thus boco/bX -- 3«( m) j3 X, and a
measurement of the absorption constant of the
induced first order infrared absorption band should provide quantitative information about the coef- ficients which determine the intensity of first order
scattering (5).
The strength of the field induced first order infrared absorption band in diamond-type crystals
is determined by the factor ei. 03BC1 = el. a1 eo Eo
where ei is the (unit) polarization vector of the
electronic field of the infrared radiation and eo is the unit polarization vector of the applied electric
field. An absorption band can be observed only
when the varying induced dipole moment, (5) Although higher order terms in the electronic polari- zability and anharmonic coupling effects will generally be small, it may be possible, under optimum conditions, to observe a field induced second order infrared absorption spectrum whose strength will be related to the matrix elements for second order Raman scattering.
03BC1= «1 eo Eo, has a component along the direc-
tion of polarization of the induced radiation. In
addition, one has the requirement of energy and momentum conservation. Thus, the selection rules are determined by the first order term in the electronic polarizability and are therefore similar
to those governing the first order Raman effect.
We may accordingly expect the (q = k sw 0) longi-
tudinal as well as the (q == k -- 0) transverse opti-
cal modes to contribute to the induced absorption,
and that the absorption bands will exhibit well-
defined polarization effects which depend on the
direction of the incident radiation and the orien- tation of the crystal and that of the applied field.
It is of interest to note that electric field induced infrared absorption bands in homonuclear mole- cules (6) were predicted by Condon in 1932 [2] and experimentally observed for H2 molecules at fields
of 105 volts/cm by Crawford and Dagg in 1953 [3].
The field induced absorption data obtained by
Crawford and MacDonald in 1958 [4] actually pro- vide the most reliable values for the polarizability
matrix elements of H 2 (7).
It should be pointed out that field induced infrared absorption bands associated with Raman active modes of vibration; should also occur for other crystal structures having a center of sym-
metry, and for impurity modes. In the case of polar crystals such as CaF2 which exhibit a first order Raman active mode, the strength of the
induced infrared absorption band will be deter-
mined by contributions from the atomic as well as
from the electronic polarizability. Since the strength of the second order infrared absorption
bands in polar crystals is appreciably greater than
in diamond-type crystals, it will be somewhat
more difflcult to observe the induced absorption
bands in polar crystals unless the induced bands fall in wavelength regions where there are no
major contributions from second order processes.
Acknowledgements. - We wish to thank Pro-
fessor J. Birman and Professor A. Maradudin for valuable discussions of electric field induced pheno-
mena.
(6) We are grateful to H. L. Welsh for informing us
about the theoretical and experimental work on the electric
field effects in homonuclear molecules.
(7) In the case of the homonuclear molecules, the rota- tional, as well as vibrational, transitions of the molecules lead to electric field induced absorption bands.
REFERENCES [1] LAX (M.) and BURSTEIN (E.), Phys. Rev., 1955, 97, 39.
[2] CONDON (E. U.), Phys. Rev., 1939, 41, 759.
[3] CRAWFORD (M. F.) and DAGG (I. R.), Phys. Rev., 1953, 91, 1569.
[4] CRAWFORD (M. F.) and MAcDONALD (R. E.), Can. J.
Phys., 1958, 36, 1022.
